1. The problem statement, all variables and given/known data Describe how to determine whether an equilibrium is stable or unstable when [d2U/dx2]_0 = 0 From Classical Dynamics - Ch 2 #45 - Marion Thornton 2. Relevant equations AND 3. The attempt at a solution When second derivative positive, equilibrium is stable. When second derivative negative, equilibrium unstable. When second derivative is 0, you have to do another test. I was under the impression that you could find a third derivative, fourth derivative, until you get a positive/negative value for x=0 and use the results to get stability, with the same rule as above? And if second and subsequent derivatives are 0, it's a neutral equilibrium. The main reason I'm confused is the book had a weird solution that didn't make sense to me. They expanded the potential about the equilibrium point (how? why?), found the first derivative, and said that it's stable when n is even and unstable when n is odd. But that doesn't make sense to me. How could they make a conclusion like that if they don't know what U is? Maybe I'm misunderstanding the question's notation - I understood it to mean "second derivative at x=0 is 0."